Question

Let utility be defined as by U(x,y)=xy+y with a budget constraint of I=pyY+pxX. Your solutions for x^* and y^* will include p_x,p_y and I.

a) Derive the Marshallian demand for x and

b) Find the income elasticity, and both price elasticities (own, cross) for x and y. (hint: it can be helpful for simplification and interpretation, once you have derived your elasticity, to substitute in your values for x or y that we derived in part a) above)

c) Contrast your answers for income elasticity for x and y. Do you get a greater than or less than 1% change in consumption, for each good, when there is a 1% increase in income?

d) What do you notice about the cross-price elasticities for each good?

Answer 1

(a) Utility is maximized when MRS = MUx/MUy = px/py

MUx = U/x = y

MUy = U/y = x + 1

MRS = y / (x + 1) = px/py

y = (x + 1).(px/py)

Substituting in budget line,

I = x.px + y.py

I = x.px + [(x + 1).(px/py)].py

I = x.px + (x + 1).px

I = x.px + x.px + px

I = 2x.px + px

2x.px = I - px

x = (I - px) / 2px [Demand function for x]

y = {[(I - px) / 2px] +1}.(px/py}

y = [(I - px + 2px) / 2px).(px/py)

y = (I + px) / py [Demand function for y]

(b)

(i) Income elasticity of x = (x/I).(I/x) = [1 / (2px)].[I / {(I - px) / 2px}] = I / (I - px)

(ii) Income elasticity of y = (x/I).(I/x) = [1 / (py)].[I / {(I + px) / py}] = I / (I + px)

x = (I - px) / 2px = (I/2px) - (1/2) = [(I.px^-1)/2] - (1/2)

y = (I + px) / 2py = (I/2py) + (px/2py) = [(I.py^-1)/2] + (px/2py)

(iii) Own price elasticity of x = (x/px).(px/x) = [(1/2) x (-1) x (I.px^-2)] = -I / (2px²)

(iv) Own price elasticity of y = (y/py).(py/y) = [(1/2) x (-1) x (I.py^-2)] = -I / (2py²)

(v) Cross price elasticity between x and y = (x/py).(py/x) = 0 (since demand function for x is independent of py)

(vi) Cross price elasticity between y and x = (y/px).(px/y) = (1/2py).{px/[(I.py^-1)/2] + (px/2py)}

(c)

For good x, when income increases by 1%, numerator increases more than the denominator, so income elasticity is higher than 1.

For good y, when income increases by 1%, denominator increases more than the numerator, so income elasticity is lower than 1.

(d)

When price of good y changes, there is no change in demand for good x since Cross price elasticity between x and y is zero. But when price of good y increases, Cross price elasticity between y and x being positive, demand for good x increases, signifying y and x are substitutes.